I have used a few polishing kits before (the kind generally sold by e.g. Halfords). The ones you can attach to a drill can save you a lot of work and time, but the manual ones work well too, as long as you are willing to put in the work (depending on how hazy the lights are, it can take a couple of hours if manually polishing). The issue, particularly for older headlights, where the lacquer coat has worn out partially or fully, is that the headlights get foggy again quite quickly. Once that protective coat has come off, you will have to polish the lenses quite regularly, especially in the winter.
Ideally, you need to re-seal the lenses again after you polish them. Even this doesn't last forever, but it will definitely keep them clear for longer.
I would strongly advise against "DIY" methods, like using toothpaste. At best, they won't do much, at worst they could cause deeper scratches or damage/yellow the plastic.
